en>Glrx: /* References */

2014-12-18T20:10:19Z

References

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	~~{{Merge\|electron density\|date=September 2010\|discuss=talk:electron density}}~~		Wilber Berryhill is what his wife loves to call him and he completely enjoys this title. Credit authorising is where my main earnings comes from. My spouse doesn't like it the way I do but what I really like performing is caving but I don't have the time recently. North Carolina is the place he loves most but now he is considering other options.<br><br>Check out my web page; [http://m-card.co.kr/xe/mcard_2013_promote01/29877 real psychic]
	In [[quantum mechanics]], and in particular [[quantum chemistry]], the '''electronic density''' is a measure of the probability of an [[electron]] occupying an infinitesimal element of space surrounding any given point. It is a scalar quantity depending upon three spatial variables and is typically denoted as either ''ρ''('''r''') or ''n''('''r'''). The density is determined, through definition, by the normalized ''N''-electron [[wavefunction]] which itself depends upon 4''N'' variables (3''N'' spatial and ''N'' [[Spin (physics)\|spin]] coordinates). Conversely, the density determines the wave function modulo a phase factor, providing the formal foundation of [[density functional theory]].

	~~==Definition==~~
	~~The electronic density corresponding~~ to ~~a normalized ''N''-electron [[wavefunction]] (with '''r'''~~ and ~~''s'' denoting spatial and spin variables respectively) is defined as<ref>{{cite book\|last1=Parr\|first1=Robert G.\|last2=Yang \| first2= Weitao\|~~title~~=Density-Functional Theory of Atoms and Molecules\|publisher=Oxford University Press\|location=New York\|year=1989\|isbn=0-19-509276-7}}</ref>~~

	~~:<math>~~
	~~\begin{align}~~
	~~\rho(\mathbf{r})&=N\sum_{{s}_{1}} \cdots \sum_{{s}_{N}} \int \ \mathrm{d}\mathbf{r}_2 \ \cdots \int\ \mathrm{d}\mathbf{r}_N \ \|\Psi(\mathbf{r},s_{1},\mathbf{r}_{2},s_{2},~~.~~..,\mathbf{r}_{N},s_{N})\|^2, \\~~
	~~&= \langle\Psi\|\hat{\rho}(\mathbf{r})\|\Psi\rangle,~~
	~~\end{align}~~
	~~</math>~~

	where ~~the operator corresponding to the density observable is~~

	~~:<math>\hat{\rho}(\mathbf{r}) = \sum_{i=1}^{N}\sum_{s_{i}}\ \delta(\mathbf{r}-\mathbf{r}_{i})~~.~~</math>~~

	~~In [[Hartree-Fock]] and [[density functional theory\|density functional]] theories the wave function is typically represented as a single [[Slater determinant]] constructed from~~ '~~'N'' orbitals, ''φ''<sub>''k''</sub>, with corresponding occupations ''n''<sub>''k''</sub>. In these situations~~ the ~~density simplifies to~~

	~~:<math>\rho(\mathbf{r})=\sum_{k=1}^N n_{k}\|\varphi_k(\mathbf{r})\|^2.</math>~~

	~~==General Properties==~~

	~~From its definition, the electron density~~ is a non-negative function integrating to the total number of electrons. Further, for a system with kinetic energy ''T'', the density satisfies the inequalities<ref name="lieb83">{{cite journal\|last=Lieb\|first=Elliott H.\|year=1983\|journal=International Journal of Quantum Chemistry\|volume=24\|issue=3\|pages=243–277\|title=Density functionals for coulomb systems\|doi=10.1002/qua.560240302}}</ref>

	~~:<math>\frac{1}{2}\int\mathrm{d}\mathbf{r}\ \big(\nabla\sqrt{\rho(\mathbf{r})}\big)^{2} \leq T.</math>~~

	~~:<math>\frac{3}{2}\left(\frac{\pi}{2}\right)^{4/3}\left(\int\mathrm{d}\mathbf{r}\ \rho^{3}(\mathbf{r})\right)^{1/3} \leq T.</math>~~

	For finite kinetic energies, the first (stronger) inequality places the square root of the density in the [[Sobolev space]] ''H''<sup>1</sup>('''R'''<sup>3</sup>). Together with the normalization and non-negativity this defines a space containing physically acceptable densities as

	~~:<math>~~
	~~\mathcal{J}_{N} =~~
	~~\left\{ \rho \left\| \rho(\mathbf{r})\geq 0,\~~
	~~\rho^{1/2}(\mathbf{r})\in H^{1}(\mathbf{R}^{3}),\~~
	~~\int\mathrm{d}\mathbf{r}\ \rho(\mathbf{r}) = N~~
	~~\right.\right\}.~~
	~~</math>~~

	~~The second inequality places the density in the [[Lp space\|''L'~~'~~<sup>3</sup> space]]. Together with the normalization property places acceptable densities within~~ the ~~intersection of ''L''<sup>1</sup> and ''L''<sup>3</sup> – a superset of <math>\mathcal{J}_{N}</math>~~.

	~~==Topology==~~

	~~The [[ground state]] electronic density of an [[atom]]~~ is ~~conjectured to be a [[Monotonic function\|monotonically]] decaying function of the distance from~~ the [[atomic nucleus\|nucleus]].<ref>{{cite journal\|last1=Ayers\|first1=Paul W.\|last2=Parr \| first2= Robert G.\|year=2003\|title=Sufficient condition for monotonic electron density decay in many-electron systems\|journal=International Journal of Quantum Chemistry\|volume=95\|issue=6\|pages=877–881\|doi=10.1002/qua.10622}}</ref>

	~~===Nuclear cusp condition===~~

	~~The electronic density displays cusps at each nucleus in a molecule as a result of the unbounded electron-nucleus Coulomb potential. This behavior~~ is quantified by the Kato cusp condition formulated in terms of the spherically averaged density, <math>\bar{\rho}</math>, about any given nucleus as<ref>{{cite journal\|last=Kato\|first=Tosio \|year=1957\|title=On the eigenfunctions of many-particle systems in quantum mechanics\|journal=Communications on Pure and Applied Mathematics\|volume=10\|issue=2\|pages=151–177\|doi=10.1002/cpa.~~3160100201}}~~<~~/ref~~>

	:<~~math~~>~~\left.\frac{\partial}{\partial r_{\alpha}}\bar{\rho}(r_{\alpha})\right\|_{r_{\alpha}=0} = -2Z_{\alpha}\bar{\rho}(0).</math>~~

	~~That is, the radial derivative of the spherically averaged density, evaluated at any nucleus, is equal to twice the density at that nucleus multiplied by the negative of the [~~[~~atomic number]] (''Z'').~~

	~~===Asymptotic behavior===~~

	~~The nuclear cusp condition provides the near-nuclear (small ''r'') density behavior as~~

	:~~<math>\rho(r) \sim e^{-2Z_{\alpha}r}\,.<~~/~~math>~~

	The long-range (large ''r'') behavior of the density is also known, taking the form<ref>{{cite journal\|last1=Morrell\|first1=Marilyn M.\|last2=Parr\|first2=Robert. G.\|last3=Levy\|first3=Mel\|year=1975\|title=Calculation of ionization potentials from density matrices and natural functions, and the long-range behavior of natural orbitals and electron density\|journal=Journal of Chemical Physics\|volume=62\|issue=2\|pages=549–554\|doi=10.1063/~~1.430509\|bibcode = 1975JChPh..62..549M }}</ref>~~

	~~:<math>\rho(r) \sim e^{~~-~~2\sqrt{2\mathrm{I}}r}\,~~.~~</math>~~

	~~where I is the [[ionization energy]] of the system.~~

	~~==Response Density==~~

	~~Another more-general definition of a density is the "linear-response density".<ref>~~
	~~{{cite journal \|doi = 10~~.~~1063~~/1.447489 \|title = On the evaluation of analytic energy derivatives for correlated wave functions \|year = 1984 \|last1 = Handy \|first1 = Nicholas C. \|last2 = Schaefer \|first2 = Henry F. \|journal = The Journal of Chemical Physics \|volume = 81 \|pages = 5031\|bibcode = 1984JChPh..81.5031H \|issue = 11 }}</ref><ref>{{ cite journal \| doi = 10.1021/j100181a030 \| title = Analysis of the effect of electron correlation on charge density distributions \| year = 1992 \| last1 = Wiberg \| first1 = Kenneth B. \| last2 = Hadad \| first2 = Christopher M. \| last3 = Lepage \| first3 = Teresa J. \| last4 = Breneman \| first4 = Curt M. \| last5 = Frisch \| first5 = Michael J. \| journal = The Journal of Physical Chemistry \| volume = 96 \| pages = 671 \| issue = 2}}</ref> This is the density that when contracted
	~~with any spin-free, one-electron operator yields the associated property defined as the derivative of the energy.~~
	~~For example, a dipole moment is the derivative of the energy with respect to an external magnetic field and~~
	~~is not the expectation value of the operator over the wavefunction. For some theories they are the same when~~
	~~the wavefunction is converged. The occupation numbers are not limited to the range of zero to two, and therefore~~
	~~sometimes even the response density can be negative in certain regions of space.<ref>{{cite journal~~
	~~\| last1 = Gordon \| first1 = Mark S. \| last2 = Schmidt \| first2 = Michael W. \| last3 = Chaban \| first3 = Galina M.~~
	~~\| last4 = Glaesemann \| first4 = Kurt R. \| last5 = Stevens \| first5 = Walter J. \| last6 = Gonzalez \|first6 = Carlos~~
	~~\| year = 1999~~
	~~\| title = A natural orbital diagnostic for multiconfigurational character in correlated wave functions~~
	~~\| journal = J. Chem. Phys.~~
	~~\| volume = 110~~
	~~\| issue = 9~~
	~~\| pages = 4199–4207~~
	~~\| doi = 10.1063~~/~~1.478301\|bibcode = 1999JChPh.110.4199G }}~~
	</~~ref>~~

	~~==See also==~~
	*[[Probability current\|Current density]]

	~~==References==~~
	~~{{Reflist}}~~

	~~{{DEFAULTSORT:Electronic Density}}~~
	~~[[Category:Atomic physics]]~~
	~~[[Category:Quantum chemistry]]~~
	~~[[Category:Density functional theory]]~~

	~~[[de:Elektronendichte]]~~
	~~[[es:Densidad (mecánica cuántica)]]~~
	~~[[fr:Densité électronique]]~~
	~~[[it:Densità elettronica]]~~
	~~[[pl:Gęstość elektronowa]]~~
	~~[[ru:Электронная плотность]~~]

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